Question: Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{a^2 - 49}{a + 7}$
Solution: First factor the polynomial in the numerator. The numerator is in the form ${a^2} - {b^2}$ , which is a difference of two squares so we can factor it as $({a} + {b})({a} - {b})$ $ a = a$ $ b = \sqrt{49} = 7$ So we can rewrite the expression as: $q = \dfrac{({a} + {7})({a} {-7})} {a + 7} $ We can divide the numerator and denominator by $(a + 7)$ on condition that $a \neq -7$ Therefore $q = a - 7; a \neq -7$